Lie Algebras

107 Pages·1970·1.568 MB·English

by  Ian Stewart (auth.)

Lecture Notes ni
Mathematics
A collection of informal reports dna seminars
Edited yb .A Dold, Heidelberg dna .B Eckmann, Z0rich
Series: Mathematics Institute, University of Warwick
Adviser: .D .B .A Epstein
721
nal Stewart
University of Warwick, Coventry/England
eiL Algebras
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galreV-regnirpS
Berlin. Heidelberg. New York 1970
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PREFACE
These notes are based on an M.Sc. course which I gave at Warwick
during the Autumn of 1969. The material divides into two largely
disjoint sections.
Part I is a relatively direct exposition of the classification
theorem for finite-dimensional semisimple Lie algebras over an
algebraically closed field of characteristic zero. The treatment is
entirely algebraic, with no appeal to analysis: connections with Lie
groups are not mentioned. The main source for this section is a
lecture course given at Warwick in 1966 by Prof. R.W.Carter, which
itself derived from lectures of Philip Hall at Cambridge. Virtually
all of the material in Part 1 is included with the classification
theorem in view, so that I have not explored a number of interesting
side-issues (such as L~¢i's splitting theorem, or representations of
semisimple algebras). This approach was dictated by two considerations:
time; and a desire to exhibit the bare bones of the proof without
extraneous matter.
Part 2 is drawn from two sources: a paper published by Brian
Hartley in 1966, and my Ph.D. thesis (1969). The object here is to
investigate the structure of infinite-dimensional Lie algebras in the
spirit of infinite group theory. In particular we consider the Lie
analogue of a subnormal subgroup (which we call a subideal) and
consider the connections between the subideals of a Lie algebra and
the structure of the algebra as a whole. This branch of the subject
is at an early stage of development, as is underlined by the appearance
in the text of a number of open questions.
I am grateful inspirationwise to Brian Hartley and Roger Carter,
and perspirationwise to Sue Elworthy who typed the manuscript.
Ian Stewart
CO~E~S
CheDt~r
0 Basic Definitions I
PART ONE : CLASSICAL THEORY OF FINITE DIMENSIONAL ALGEBRAS
1 Representations of Nilpotent Algebras I0
2 Caftan Subalgebras 15
3 The Killing Form 19
The Caftan Decomposition 24
5 Systems of Fundamental Roots 28
6 Dynkin Diagrams 34
7 Some Astronomical Observations D~
8 Algebras with a Given Star 48
PART TWO: INFINITE DIMENSIONAL LIE ~GEBRAS
9 Subideals, Derivations, Automorphisms 59
i0 The Baer Radical 65
Ii Other Radicals 71
12 Baer ~ Fitting 74
13 Lie Algebras in which every Subalgebra is a
Subideal 80
14 The Minimal Condition for Subideals 90
BASIC DEFINITIONS
Let k be ~ field, ~nd L ~ vector snsee over k (of f~nite or infin-
It~ d~men~ion). L is said to be a Lie algebra over ~ if there is a
billnear multiplication (x,y)* Ix,y] defined on L such tha~
I) Ix,x] = 0 for all x ~ L,
2) [[x,y],z] + [[y,z],x] + [[z,x],y] = 0 for all x,y,z ~ L.
(Jacobl identity).
Note:
l) Multiplication is not necessarily associative.
2) If we put x+y for x in (1) we find that Ix,y] = - [y,x].
Examples
l) Any vector space with [x,yl = 0.
2) An associative algebra over k is a vector space with a billnear
associative multiplication (e.g. kn, the set of all nxn matrices over
).k__ Suppose this multiplication is denoted by Ju~Ctaposition. Then if
A is any associative algebra it becomes a Lie algebra [AS on putting
[x,y] = xy,yx
(which is where the Jacobl identity comes from).
3) Any subset of an associative algebra closed under this operation
(and vector operational, e.g. matrices of trace zero.
If H, K are subspaces of L we write [H,K] for the subspsce s~nned
by all [h,k] (h~ H, k~ K), and similarly H+K.
H is a (Lie) subal~ebra of L if [H,H]_< H, and is then a Lie algeb-
ra in its own right under the same operations as L. We write H .L(_
H is an ideal of L if [H,L](H. For this we write H ~ L.
Note :
By previous note 2 left and right ideals are the same.
2
Exercise:
i) If L is the Lie algebra of nxn k-mstrlces and H is the set of
trace zero matrices, show H 4 L.
2) H,K __< L~ PhK <__ L.
H,K ~ L *- HnK @ L,
H+K m L,
H ~ L, K _< L _~ H+K ~ L.
Suppose H 4 L. Then H is a subspace of L. The cosets H+x (xEL)
can be made into the quotient algebra L/H if we define
x (H+x) - H+Xx (X~)
(H+x)+CH+y) = H+(x+y)
[H+x,H+y] = H+[x,y].
(H must be an ideal for this to be well-defined).
The map 9:L ~- L/H with x@ = H+x is a Lie homomorphism (in the
usual sense that it preserves the operations). We may also define
isomorphlsms, automorphisms, etc.
Isomorphism theorems
The standard isomorphism theorems hold, namely:
a) Suppose @:L ~- M is a homomorphism. Then L@ ._< M, ker(@) ~ L, and
~'ker (o) = L~.
b) If H ~ L, K < L, then H a H+K, HOK a K, and H+K/K --" K/HOK.
c) If H _< K, H,K ~ L, then (L/K)/(H/K) = L/H.
d) The natural map @ sets up a bijective correspondence between
ideals of L/H and ideals of L between H and L (and between subalgebras
in a similar way).
The proofs are as for groups, and are left as an exercise.
Nilpotent and so!uble Lie algebras
L is abelian if [L,L] = 0. Let~ denote the class of abelian
Lie algebras.
Define inductively I = L, L L n+l = [Ln,Lj. We can easily see that
i 4 L L. L si nilpotent if L n+l = 0 for home n. The least such n is
the class of nilpotency of L. Let ~c be the class of nilpotent
algebras of class ~ c, andS= e U ~c the class of nilpotent algebras.
Varying things a bit, define L )0( = L, L (n÷l) = CL(n),L(n)s.
Again (n) ~ L L. L is soluble if L )n( = 0 for some n. The least such
n is the derived length of L.~d, ~ are the classes of soluble
algebras of derived length ~ ,d and all soluble algebras, respectively.
The sequence L1,L ,2 ... is the lower central series of L.
(O), L L(1),... is the derived series.
Exercise
Every subalgebra or quotient algebra of a Lie algebra in
~'~ "~d' ~ is ditto.
Example
Let T be the Lie algebra of nxn O-triangular matrices over k,
that is, matrices of the form
Show that T is nilpotent, and that r T consists of matrices of the form
0 .,
with r diagonal rows of zeros.
Show that the algebra of triangular matrices is soluble.
Theorem 0~
a) [Lm, Ln] ! Lm+n-
n 2
b) L(n) ! L .
Proof:
a) Induction on n. True for n I by definition.
=
Now [Lm,Lr*I]= [[Lr,L],L m]
[[L,Lm],L r] + [[Lm,Lr],L] (Jacobi)
L m+r+l by induction.
b) Follows by induction from a), and c) from b).
Exercise
If H ~ L then n 4 H L and (n) H 4 L.
Lemma 0 ~2
a) H, L/H~ LE~ .
b) K,H q L, H,K~ H.KG~.
c) K,H 4 L, H,E~ ~ ~K+H ~ .
Proof:
a) L/H~ ~ (L/H) (d) = 0 Z L(d) ~ H. H (e) = O. Hence L (d+e) = O.
b) H+K/K ~ H/HnKE~, and KE~ . Now use part (a).
c) Suppose H c+l = 0, K d+l = 0. Consider J = (H+K) T M • Elements
of this are linear combinations of elements I, [x ... , Xc+d+ I]
where the i x lie in H or K. Either ~ c+l of them lie in H or ~ d+l in
K. By symmetry we may assume ~ c+l lle in H. By the above exercise
all powers of H are ideals of L. Thus this element lies in H c+l = 0
so J = 0 and H+K~%+ d.
We may now define the two classical radicals of a finite-dimens-
ional Lie algebra:
Theorem 0.5
Let L be a finlte-dimenslonal Lie algebra. Then L has a unique
maximal nilpotent ideal N and a unique maximal soluble ideal S. L/S
has no non-zero soluble ideals.
Proof:
Let S be the sum of all soluble ideals of L. L is finite-dimen-
sional so S is the sum of a finite number of them. By lemma 0.2 part
(b) S is soluble and by definition contains every soluble ideal of L.
For N substitute 'nilpotent' for 'soluble' and use 0.2 part (c).
Suppose L/S has a non-zero soluble ideal A/S. Then A ~ L, A ~ S.
A/S and S ~, so A ~ by 0.2.a. Contradiction.
N is the nil radical of L, S the radical. A finite-dimensional
Lie algebra L is semisimple iff it has zero radical. It is with such
algebras that the classical theory is mainly concerned. We may restate
the last part of theorem 0.3 as:
Corollary
Let L be a finite-dimensional Lie algebra, S the radical of L.
Then L/S is semisimple.
Back to the genzral case with an
Exercise
Central series.
A series of subalgebras
o 0 = L _( I L _~ ... _( n = L L
is a central series for L if [Li+l,L _(] i L for all i with 0 _( i _~ n-l.
Show that i 4 L L, that L ~n, and i L _~ L n+l-i.
The centre of L is
~l(L) = ~x~L:[x,L]-- .10
Show ~I(L) ~L.
Define the upper centra! series by ~r+l(L)/~r(L) --
~I(LNr(L)). Then ~r(L) 4L, and in the above situation ~r(L) _~ i. L
Thus every central series of L is sandwiched between the lower and
upper central series of L. Hence the names.
L is simple if its only ideals are 0 and L. The only simple
soluble algebras are 1-dimensional, and are said to be trivial simple
algebras.
Representations
Representations enable us to get at general Lie algebras via ones
more easily dealt with - namely matrix algebras (where the full power
of linear algebra is on our side).
Suppose L is a Lie algebra over k. A representat!on of degree n
of L is a homomorphism
0: L * K~n]
where [~n ] denotes the Lie algebra of all nxn k-matrices.= Thus P is
a linear map whicL also satisfies
~([x,y]) = ~(x)~(y) -~(y)~(x).
Two representations 9, o are equivalent if there is a nonsingular k-
matrix T such that for all x
o(x) = T-10(x)T.
(This corresponds to a base change in the matrices, and becomes
important if we take a more abstract view as follows:)
An L-module is a finite-dimenslonal vector space V over k admit-
=
ring a multiplication by elements of L denoted by (v,x) ~vx (v~V,xEL)
such that for all x,yEL, u,~V, kE k we have
(u+v)x = ~x+vx
(~u)x = ~(ux) = u(~x)
u(x+y) = ux+uy
u[x,y] = (ux)y- (~)x.
Given a basis el,...,e n of V, V determines a representation of L
as follows: